Problem: The dilation, centered at $-1 + 4i,$ with scale factor $-2,$ takes $2i$ to which complex number?
Answer: Let $z$ be the image of $2i$ under the dilation.

[asy]
unitsize(0.5 cm);

pair C, P, Q;

C = (-1,4);
P = (0,2);
Q = (-3,8);
draw((-5,0)--(5,0));
draw((0,-1)--(0,10));
draw(P--Q,dashed);

dot("$-1 + 4i$", C, SW);
dot("$2i$", P, E);
dot("$-3 + 8i$", Q, NW);
[/asy]

Since the dilation is centered at $-1 + 4i,$ with scale factor $-2,$
\[z - (-1 + 4i) = (-2)(2i - (-1 + 4i)).\]Solving, we find $z = \boxed{-3 + 8i}.$